Bi-infinite Riordan matrices: A matricial approach to multiplication and composition of formal Laurent series

L.F. Prieto-Martínez, J. Rico

Linear Algebra and its Applications Vol. 731, pp. 139 - 159

Resumen:

We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily lower triangular and are determined, not by a pair of formal power series, but by a pair of formal Laurent series. We extend the First Fundamental Theorem of Riordan Matrices to this setting, as well as the Toeplitz and Lagrange subgroups, that are subgroups of the classical Riordan group. Finally, as an illustrative example, we apply our approach to derive a classical combinatorial identity that cannot be proved using the techniques related to the classical Riordan group, showing that our generalization is not fruitless.

Resumen divulgativo:

Generalizamos el concepto de matrices de Riordan a un entorno bi-infinito adecuado para la multiplicación y composición de series de Laurent formales. Además, ampliamos el Primer Teorema Fundamental de las Matrices de Riordan y mostramos su uso derivando una identidad combinatoria clásica.

Palabras Clave: Formal Laurent series; Bi-infinite matrices; Riordan group

Índice de impacto JCR-JIF y cuartil WoS: 1,100 - Q2 (2025)

Referencia DOI: https://doi.org/10.1016/j.laa.2025.11.010

Publicado en papel: Febrero 2026.

Publicado on-line: Noviembre 2025.

Cita:
L.F. Prieto-Martínez, J. Rico, "Bi-infinite Riordan matrices: A matricial approach to multiplication and composition of formal Laurent series", Linear Algebra and its Applications, Vol. 731, pp. 139 - 159, Febrero 2026. [Online: Noviembre 2025] doi: 10.1016/j.laa.2025.11.010